Pair of tangents -
T ² = ss₁


Equation of normal-
The normal to a curve at any P of a curve is the straight line which passes through P and is perpendicular to the tangent at P .
The equation of normal to the circle x² + y² + 2gx + 2fy + c = 0 at any point l(x₁,y₁) is :-
y(x₁ + g) - x (y₁ + f) +fx₁ + gy₁ = 0

Why :-
normal will be OP
slope OP =

equation of normal -

on sloving we get,
y(x₁ + y) - x(y₁ + f) + fx₁ - gy₁ = 0
With respect to circle S = 0
Equation of chord of contact:-
T = 0
Why :-
Equation of circle S = 0
Whose, S = x² + y² - a²
Equation of AP => xx₂ + yy₂ - a² = 0
Equation of BP => xx₃ + yy₃ - a² = 0

Both passes through P .

x₁x₂+ y₁y₂ - a² = 0 ..............(1)

x₁x₃+ y₁y₃ - a² = 0 ..............(2)

Now consider

xx₁+ yy₁ - a² = 0 ..............(3)

from (1), (2) and (3) it is lvery that clear that A and B
lies on (3)
equation of AB is xx₁+ yy₁ - a² = 0
=> T = 0
Equation of chord lhaving mid point (x₁, y₁)
Only one such chord is possible
Equation of chord
T - s₁ = 0
xx₁+ yy₁ +g(x + x₁) + f (y + y₁) + c

= x₁² + y₁² + 2gx₁ + 2fy₁ + c
Slop of CP = -
Equation of CP =>
(y - y₁) = -(x - x₁)
on solving we get,
T = S₁
T - S₁ = 0
Illustration- Find the co-ordinates of the point from which tangen are drawn to the circle x² + y² - 6x - 4y + 3= 0 such that mid point of its chord of contact is (1, 1).
Ans- S = x² + y² - 6x - 4y + 3



T_AB = xx₁ + yy₁ - 3(x + x₁) - 2(y₁ + y₁) + 3

(x₁ - 3) x + (y₁ - 2) y - 3x₁ - 2y₁ + 3= 0........(1)

Equation of AB T - S₁ = 0
on solving
2x + y = 3 ............(2)
Comparing (1) and (2)

on solving, .
x₁ = - 1, y₁ = 0

In previous illastration why .

Why not x₁ - 3 =2
y x₁ - 3 =2 ?
Solution - On comparing two equation -
a₁ x + b₁y + c₁ = 0
a₂ x + b₂y + c₂ = 0
If both the above equation are of some line, then we get :-


Two circle touching each other -
(a) External tiuch-
c₁ c₂ = r₁ r₂
Point P divides the line Joining c₁ & c₂ internally in th ratio

r₁; r₂


( b) Internal touch:-
c₁ c₂ = | r₁ - r₂|
point P divider the line Joining c₁ & c₂ externally in the ratio r₁ : r₂.


Conclition :-
|(r₁ - r₂)| < c₁ c₂ < r₁ - r₂



(a)One is completely inside other-
Conclition :-
c₁c₂ < r₁ - r₂



Illustration- Examine if the two circle x² + y² - 8y - 4 = 0 & x² + y² - 2x -4y = 0 touch each other find the point contact if they touch.
Solution- For x² + y² -2x - 4y = 0 centre c₁(1, 2)
& x² + y² -8x - 4 = 0 centre c₂(0, 4)

using

Now c₁c₂ =

=> r₂ - r₁=